Asymptotic expansions for a class of tests for a general covariance structure under a local alternative

作者：Shimizu Hiroaki; Wakaki Hirofumi^*

来源：Journal of Multivariate Analysis, 2011, 102(6): 1080-1089.

DOI：10.1016/j.jmva.2011.03.002

• Asymptotic expansion
• Class of test statistics
• General covariance structure
• Non-null distribution
• Local alternative
• Power comparison
• Linear structure
• Sphericity test

Let S be a p x p random matrix having a Wishart distribution W-p(n, n(-1) Sigma). For testing a general covariance structure Sigma = Sigma(xi), we consider a class of test statistics T-h = n rho(h)(S, Sigma((xi) over cap)), where rho(h) (Sigma(1), Sigma(2)) = Sigma(p)(i=1) h(lambda(i)) is a distance measure from Sigma(1) to Sigma(2), lambda(i)'s are the eigenvalues of Sigma(1)Sigma(-1)(2), and his a given function with certain properties. Wakaki, Eguchi and Fujikoshi (1990) suggested this class and gave an asymptotic expansion of the null distribution of T-h. This paper gives an asymptotic expansion of the non-null distribution of T-h under a sequence of alternatives. By using results, we derive the power, and compare the power asymptotically in the class. In particular, we investigate the power of the sphericity tests.

• 出版日期2011-7